L(s) = 1 | + 1.85i·3-s + 5-s + (1.29 + 2.30i)7-s − 0.432·9-s + 2.01·11-s + 1.82·13-s + 1.85i·15-s + 1.64i·17-s + 1.26i·19-s + (−4.27 + 2.40i)21-s + 3.19i·23-s + 25-s + 4.75i·27-s − 4.57i·29-s − 0.834·31-s + ⋯ |
L(s) = 1 | + 1.06i·3-s + 0.447·5-s + (0.490 + 0.871i)7-s − 0.144·9-s + 0.608·11-s + 0.507·13-s + 0.478i·15-s + 0.399i·17-s + 0.289i·19-s + (−0.931 + 0.525i)21-s + 0.665i·23-s + 0.200·25-s + 0.915i·27-s − 0.849i·29-s − 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.279605840\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.279605840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.29 - 2.30i)T \) |
good | 3 | \( 1 - 1.85iT - 3T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 1.64iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 3.19iT - 23T^{2} \) |
| 29 | \( 1 + 4.57iT - 29T^{2} \) |
| 31 | \( 1 + 0.834T + 31T^{2} \) |
| 37 | \( 1 + 7.82iT - 37T^{2} \) |
| 41 | \( 1 - 2.75iT - 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.65iT - 53T^{2} \) |
| 59 | \( 1 + 3.88iT - 59T^{2} \) |
| 61 | \( 1 + 0.285T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 8.39iT - 89T^{2} \) |
| 97 | \( 1 + 9.32iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.330313448085925012113710222243, −8.727728645424814937243871588583, −7.898005211617903357886818348183, −6.84686187703753452628502186097, −5.79413477056020809471336496100, −5.43572532170520276551121992581, −4.30130802644940327770047079733, −3.76620379595144538132977783797, −2.52409512212515903717410031555, −1.44595599307493927504803025962,
0.888713078232929315138416883519, 1.59558678090460161598109450184, 2.72405684185592092593576047986, 3.97368911485162551427976586298, 4.76734524190549696405472801019, 5.88604874372245262738438093853, 6.64609593355150942704806933254, 7.19783273653774853465230322801, 7.87175547588161611330760860283, 8.746824160883701996094453662523